As means for reducing the exhaust from, for example, a diesel engine or the like, there is known technology of using a catalyst (hereinafter referred to as a catalyst (diesel particulate filter: hereafter it's called as “DPF”) that captures soot and purifies the same, in particular, contained in exhaust. The DPF captures soot contained in the exhaust from the engine during normal operation of the engine. When the DPF is to be purified, the exhaust temperature is raised (preferably to 600° C. or higher) and the soot contained in the exhaust is burned by an oxidant (oxygen) contained in the exhaust. In order to efficiently purify the soot contained in the exhaust, it is a key point to accurately grasp the quantity of exhaust heat and the concentration of oxygen to be provided for the DPF to control the exhaust heat quantity and the oxygen concentration. In technology disclosed in patent document 1, for example, the exhaust heat quantity is estimated based on the output value of an intake air flow rate sensor and the quantity of fuel injection (quantity given by a control command) and, a catalyst such as a DPF or the like is controlled based on the estimated exhaust heat quantity.